Here goes for the quick starter answers:
1) DFTI (distance from touchdown indicator) has been mostly replaced by ATM (air traffic monitor), which can be as good as the radar used by approach controllers.
2) Theory of what to do with an ATM is covered in the aerodrome course. How it works used to be covered in Radar Theory later on. This will change very soon. Each ATM is different, so functions picked up during OJT.
3) Range varies with equipment, but not unusual to be able to wind out beyond 40nm for spotting, or even operational planning. Normally it would be 15-20nm range. Most functions available to the tower controller and no radar rating required as it is usually a monitoring function only (I know 'advanced use of ATM' etc.). Radar rating is necessary for radar position only.
4) Various units provide approach procedural service from the Visual Control Room (eg Woodvale, Shoreham, Oxford, Cranfield etc), but I am unaware of any which have approach radar in the VCR (historically very difficult to see the radar screen in daylight). It is also specifically forbidden to provide both a Tower and Approach Radar service simultaneously.